Since the discovery of the alpha rhythm, rhythmic activity has been experimentally observed in many parts of the central nervous systems (CNS) of living organisms. Such activity can be traced down to the single cell level where the rhythmicity is associated with membrane potential oscillations at different frequencies and subserved by different ionic mechanisms. Indeed, this kind of activity is supported by the biophysical properties of single neurons and often reinforced by their connectivity.
Various feedback and feed-forward resonance loops “linking” different parts of the CNS select a particular rhythm. An important source of input connectivity to the cerebellum is the inferior olive (IO) nucleus. The olivo-cerebellar circuit plays an important role in motor performance and the control of movement. In this circuit, the inferior olive projects excitatory signals into the Purkinje cell layer in the cerebellar cortex. In turn, the Purkinje cells send inhibitory messages to the IO via the cerebellar nuclei which also receive excitatory inputs from the IO. The inferior olive is believed to act as a quasi-digital timing device for movement coordination functions of the CNS.
FIGS. 1A-C show the main functional structures of the olivo-cerebellar circuit and their interconnectivity. The Purkinje cells (PCs) provide feed-forward inhibitory control to the cerebellar nuclei (CN). The CN have two distinct functions: 1) to provide inhibitory feed-back to IO neurons to control neuron coupling in the IO; and 2) to provide inhibitory feedback to a second set of neurons that have excitatory termination in the thalamus, the brain stem, and the upper spinal cord. This second set of neurons times motor execution. Thus, in the olivo-cerebellar circuit, the PCs modulate (via the nuclear feed-back inhibitory pathway) cluster formation in the inferior olive, and inhibit excitatory output to the rest of the brain.
The inferior olive nucleus consists of neurons that are capable of supporting rhythmic symmetric membrane potential dynamics with respect to the base line (i.e. the rest potential). These almost sinusoidal sub-threshold oscillations support spike generation when the membrane is depolarized or hyperpolarized (sodium and calcium channels, respectively). Precision and robustness of spike generation patterns primarily result from the precision and robustness of the sub-threshold oscillations.
As shown by in vitro and in vivo experiments, the behavior of IO neurons exhibits two characteristic features: (i) spontaneous sub-threshold oscillations having an amplitude of 5 to 10 mV, with a quasisinusoidal shape and a frequency of 5 to 12 Hz, as shown in FIG. 2B; and (ii) action potentials or spikes at the top of the oscillations when the threshold is exceeded, as shown in FIGS. 2A and 2C.
Presently only a few models of the IO neuron have been developed, such as the Manor, Rinzel, Yarom and Segev model and the Schweighofer, Doya and Kawato model. These models primarily address the dynamics of neuronal integration by detailed ionic conductance and passive multicompartment cable modeling. Such models may be tuned according to particular experimental conditions and may describe the behavior of the IO neuron very accurately. For purposes of a possible implementation with electronic circuitry, however, these models are quite complex involving a huge number of variables and parameters.
Another approach to modeling the IO neuron is behavior-based modeling. Examples are the FitzHugh-Nagumo two-variable model describing the propagation of action potentials and the Hind-March-Rose model providing spike-burst oscillations. Such models exhibit good qualitative agreement with actual neuron behavior and are simpler to implement electronically.
In addition to studying the behavior of individual IO neurons, significant work has been done in observing and characterizing the behavior of large numbers of interacting IO neurons, as they exist in the inferior olive. In the actual inferior olive, the neurons are coupled to each other via gap junctions. As has been observed experimentally, interactions between IO neurons is largely local; i.e., ensembles of interacting neurons tend to involve a relatively small number of neighboring neurons.
As the coupling among neurons increases, the degree of synchronization of the neurons increases. As has been experimentally observed, closely coupled IO neurons form oscillatory clusters. Influencing such synchronism and intercoupling of neurons is the olivo-cerebellar loop. Via certain pathways, the cerebellum can influence the degree of coupling among IO neurons, thereby increasing or decreasing the degree of synchronism among IO neurons. This feedback mechanism leads to the formation of patterns of neuron clusters in the IO having both a temporal and a spatial distribution.
On the one hand, IO neurons can behave as autonomous oscillators and on the other as a neuronal ensemble producing synchronous spikes. The neuron action potentials are transmitted over axons to the Purkinje Cell array (PC) and the cerebellar nuclei (CN). The CN neurons return to the IO as inhibitory terminals situated mostly on the gap junctions to implement the electrical decoupling between the IO neurons. This return pathway serves as a feedback inhibitory, decoupling signal to the IO neurons, creating conditions for multi-cluster activity.
The dendrites of the closely packed IO neurons are electrotonically coupled via dendritic gap junctions which serve to synchrone their oscillatory properties. Primary coupling occurs between 50 or so neighboring cells. In addition to activating Purkinje cells, IO spikes activate the inhibitory cerebellar nucleus loop which projects to the IO glomeruli where the gap junctions occur and produces a dynamic shunt of the electrotonic coupling. These two mechanisms lead to internal synchronization and desynchronization, which together with sensory and motor input result in the formation of spatio-temporal oscillatory activity clusters in the IO. Such clusters have been studied using voltage dependent dye imaging of the IO in vitro and have been monitored in vivo at the Purkinje cell layer with multiple electrode recordings. The time appropriate dynamics in the clusters' activity have been directly correlated with pre-motor patterns of Purkinje cell activity during motor execution.
Multi-electrode experiments with Purkinje cells in the rodent cerebellar cortex have also shown that the number of cells producing isochronous spike clusters is relatively small for spontaneous activity and increases with neuropharmacological intervention with drugs such as harmaline (hyperpolarizes further IO neurons) or picrotoxin (prevents decoupling of gap junctions). In the latter case, almost all neurons are grouped into one cluster and fire together. This reentry provides a means for an external stimulus to control the sensitivity of the loops IO-PC-CN and IO-CN. Such modulations allow the formation of well-organized patterns of global activity, which are of significance in motor coordination. The patterns evolve in time as the autonomous excitatory-inhibitory loops suitable recognize the clusters of synchronous firing neurons and prevent their uncontrolled growth. The clusters in the IO generally reorganize as the amplitudes decay with subsequent phase resetting.
It is believed that the spatio-temporal clustering of IO neurons is used to select the optimal combination of simultaneous muscular contractions to carry out coordinated movements by acting as movement execution templates. The clustering is updated at a rate of approximately 10 Hz, the frequency of sub-threshold oscillations of the IO neurons.
Operating with oscillatory space-time dynamics, the olivo-cerebellar system serves as an effective analog controller with surprisingly high computational power. Unlike a digital system, the olivo-cerebellar system does not actually perform computations, but rather deals with analog signals and represents the parameters under control as space-time patterns.
This system can provide simultaneous, on-line tuning of a large number of parameters (e.g., muscular parameters) with the precision required to execute the complex multi-jointed movements that characterize vertebrate motricity. For instance, a simple grasping movement of a hand involves the simultaneous activation of 50 key muscles with more than 1015 possible combinations of contractions. By comparison, a digital controller updating parameters every 1 ms would require a clock rate on the order of 106 GHz. A digital solution would likely entail an independent processor for each muscle via a parallel digital controller. However, activation of different groups of muscles (muscle synergies) should be highly coherent and at each time step the processors would require highly precise synchronization. It is thus apparent that even simple motor tasks would result in computational overload of conventional processors.
By contrast, the olivo-cerebellar system operates with a drastically different strategy. First, in order to avoid the huge computational workload, the olivo-cerebellar system operates in a temporally discontinuous fashion. The IO operates at approximately 10 Hz, which appears as a physiological tremor and results in the discontinuity of movement. At the same time, the low timing rate demands recurrent upgrade compensation every 100 msec to smooth the movement discontinuities. As discussed, this is implemented through dynamic nucleo-cerebellar inhibitory feedback on IO oscillatory phase by changing the electrotonic coupling among IO neurons.
Movement control requires that each time step activation of different muscles or muscle synergies be highly synchronized. Accordingly, the IO neurons, which act as controller oscillators, form a set of phase clusters with spatial configuration corresponding to the muscle contraction template. Thus, the space-time evolution of the clusters controls the optimal template at the next time step. Note, that such an internal representation of the parameters under control brings a high degree of resilience to the system. Indeed, if one of the parameters (or a control unit) is damaged, the IO can rapidly rearrange cluster distribution and execute the required action.